Recently, I’m discovering the forming reason of quantum mechanics. In the beginning of this journey, I encountered a more fundamental question from electromagnetism:
Why do accelerated charges radiate?

I attempted to find a intuitive explanation as the first step, searching for the answer on Reddit, StackExchange and Wikipedia.

And I do found some helpful resources, including a talking about the problem:

and even a brand new concept:

As I know that moving charges generate a relative static electric field and magnetic field, the question haunting me is why do accelerated charges radiate instead of a uniformly moving charge?
The change of electric field of a accelerated charge spreads outwards at the speed of light, forming a shell of “kink” in the field, but for uniformly moving charge, the electric field also changes and spreads at the speed of light, why does only accelerated one radiates?

This introduces another question: what does the field of a uniformly moving charge look like? It seems to be smooth and no kinks.

Deeper Understanding

We can always dive deeper by virtue of mathematics. The electric field generated by a static charge is

E(r,t)=14πϵ0qr2r^\mathbf{E}(\mathbf{r}, t) = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{\mathbf{r}}

and for a uniformly moving charge, it is smooth and continuous. Suddenly I realized that the distinction between the two is just like the forming condition of shock waves.

A uniformly moving charge can generate a uniformly changing wave like the Doppler Effect, which means that the change of field is mathematically continuous. This kind of change is not radiation, so what accelerated charges generate is something else that breaks the continuity.

But the kink must be smooth without sharp corners, just the same as the uniformly moving one. So what makes it radiate is most likely to be some deeper geometric consistency.

Braking Radiation (Bremsstrahlung)

However, the term is frequently used in the more narrow sense of radiation produced when electrons (from whatever source) decelerate in matter.

Braking Radiation

Synchrotron radiation

Synchrotron radiation (also known as magnetobremsstrahlung) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (a ⊥ v).

Synchrotron Radiation

Cyclotron radiation

In particle physics, cyclotron radiation is electromagnetic radiation emitted by non-relativistic accelerating charged particles deflected by a magnetic field.

Cyclotron Radiation

Beta Decay

In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle (fast energetic electron or positron), transforming into an isobar of that nuclide.

Beta Decay

Cherenkov Radiation

Liénard–Wiechert potential

Rindler coordinates